Burgers Turbulence
Velocity Spectrum Let us consider a fluid stirred so vigorously that the stirring action creates shocks. In order to get the power spectrum of the ensuing turbulence we examine the Fourier transform of the spatial profile of the kinetic energy per unit mass (i.e. velocity squared) for a single shock. That spatial profile has the form e \left(x\right) \approx v^2 \left( x \right ) \propto \theta \left(x \right ) where \theta \left( x \right) is the Heaviside step function. Its Fourier transform is \tilde{e} \left( k \right) \propto \int_{-\infty}^{\infty} \theta \left( x \right) \exp \left(i k x \right) dx \propto \frac{1}{k} The power spectrum is therefore proportional to P \propto \frac{e}{k} \approx \frac{v^2}{k} \approx k^{-2} The spectrum deviates from this form when the width of the shock becomes comparable to the wavelength. For example, if the fluid is not ideal, but has some viscosity \nu , then the upper limit on the wavenumber k is k < v_0/\nu where v_0 is the velocity of the shock. Density Distribution From the previous section we know that the velocity scales with wavelength as v \propto l^{1/2} . In steady state we assume a constant energy transfer rate between different scales so \rho v^3/l is constant, and therefore \rho \propto l^{-1/2} . If we consider a sphere of radius l , the mass inside it would be m \propto \rho l^3 \propto l^{5/2} . This scaling relation is not too different from observed distribution of molecular clouds (see Kritsuk et al 2007). Near Shock Waves Near shock waves we get a different density distribution. Let us consider the Burgers equation \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0 Let u_0 \left(a\right) be initial velocity profile and a is the initial position of a fluid element. Initially, each fluid element continues to move at it's initial velocity. The trajectories of different fluid elements can intersect, and at those intersection points shocks can form. Before shocks form, the implicit solution to the Burgers equation with given initial conditions is given by x = a + t u_0 \left(a\right) where x is the position of the particle at time t . Let us consider the evolution near the place where a shock wave forms. Without loss of generality, let's assume that the shock forms at a= 0 at time t_s . Assuming that the initial velocity distribution is continuous, it can be expressed in terms of a Taylor expansion u_0 \left(a \right) \approx -\frac{a}{t_s} + \frac{\xi}{6} a^3 We note that we have the freedom to choose a reference frame, so we can always choose one where the constant term disappears. Also, We note that the shock forms first at the point with the largest velocity gradient, and so the quadratic term must vanish. x \approx \left(1-\frac{t}{t_s} \right) a + \frac{t \xi}{6} a^3 Just before the formation of the shock t_s - t \ll t_s the relation between the current and initial position is given by x \propto a^3 . We now turn to the calculation of the density, assuming an initially uniform density distribution \rho_0 . From the conservation of mass, we can get the density at a later time \rho dx = \rho_0 da \Rightarrow \frac{\rho}{\rho_0} = \left(\frac{d x}{d a}\right)^{-1} = \left+ \frac{t \xi}{6} a^2\right^{-1} In order for the density to exceed a certain values, two conditions must be satisfied, the first is on the time \frac{t_s-t}{t_s} < \frac{\rho_0}{\rho} Hence, finite densities much larger than the initila densities \rho_0/\rho \ll 1 only exist very close to the shock formation time. The second condition is on the location \frac{t_s \xi}{6} a^2 < \frac{\rho_0}{\rho} The probability of probing the vicinity of the shock at some random place and time and measureing a density in excess of \rho scales with critical length x and time \Delta t = t_s - t so P_{>} \left(\rho\right) \propto x \Delta t \propto a^{1/3} \Delta t \propto \left(\frac{\rho_0}{\rho}\right)^{-5/2} Category:Hydrodynamics